Inversion of normal incidence seismograms

Abstract

The well‐known instability of Kunetz’s (1963) inversion algorithm can be explained by the progressive manner in which the calculations are done (descending from the surface) and by the fact that completely different impedances can yield indistinguishable synthetic seismograms. Those difficulties can be overcome by using an iterative algorithm for the inversion of the one‐dimensional (1-D) wave equation, together with a stabilizing constraint on the sums of the jumps of the desired impedance. For computational efficiency, the synthetic seismogram is computed by the method of characteristics, and the gradient of the error criterion is computed by optimal control techniques (adjoint state equation). The numerical results on simulated data confirm the expected stability of the algorithm in the presence of measurement noise (tests include noise levels of 50 percent). The inversion of two field sections demonstrates the practical feasibility of the method and the importance of taking into account all internal as well as external multiple reflections. Reflection coefficients obtained by this method show an excellent agreement with well‐log data in a case where standard estimation techniques [deconvolution of common‐depth‐point (CDP) stacked and normal‐moveout (NMO) correction section] failed.